Sharpness of the V_g-based upper bound for top-degree graph cohomology

Prove that, for the choice of subspace V_g defined as the span of the barrel-like generators X_π and Y_π (equation (20)), the dimension bound from Corollary 3.2—computed as dim B_g − rank(d) + dim B_g^⊥—is sharp, i.e., equals the true dimension of H^{top}(GC_n^{g-loop}).

Background

The authors extend Kneissler’s method to even and odd n, reduce top-degree classes to barrel diagrams B_π, and derive an explicit, computable upper bound on H^{top}(GC_n^{g-loop}) using a chosen subspace V_g and a complementary subspace B_g^⊥.

Extensive computations show the upper bound is sharp in all cases they could check (and matches known lower bounds for odd n). Motivated by this evidence, they conjecture the bound is always sharp.